Institute for Mathematical Sciences

Preprint ims01-07

F. P. Gardiner, J. Hu and N. Lakic
Earthquake Curves.

Abstract: The first two parts of this paper concern homeomorphisms of the circle, their associated earthquakes, earthquake laminations and shearing measures. We prove a finite version of Thurston's earthquake theorem \cite{Thurston4} and show that it implies the existence of an earthquake realizing any homeomorphism. Our approach gives an effective way to compute the lamination. We then show how to recover the earthquake from the measure, and give examples to show that locally finite measures on given laminations do not necessarily yield homeomorphisms. One of them also presents an example of a lamination ${\cal L}$ and a measure $\sigma $ such that the corresponding mapping $h_{\sigma}$ is not a homeomorphism of the circle but $h_{2\sigma}$ is. The third part of the paper concerns the dependence between the norm $||\sigma ||_{Th}$ of a measure $\sigma$ and the norm $||h||_{cr}$ of its corresponding quasisymmetric circle homeomorphism $h_{\sigma}$. We first show that $||\sigma ||_{Th}$ is bounded by a constant multiple of $||h||_{cr}$. Conversely, we show for any $C_0>0$, there exists a constant $C>0$ depending on $C_0$ such that for any $\sigma $, if $||\sigma ||_{Th}\le C_0$ then $||h||_{cr}\le C||\sigma ||_{Th}$. The fourth part of the paper concerns the differentiability of the earthquake curve $h_{t\sigma }, t\ge 0,$ on the parameter $t$. We show that for any locally finite measure $\sigma $, $h_{t\sigma }$ satisfies the nonautonomous ordinary differential equation $$\frac{d}{dt} h_{t\sigma}(x)=V_t(h_{t\sigma}(x)), \ t\ge 0,$$ at any point $x$ on the boundary of a stratum of the lamination corresponding to the measure $\sigma.$ We also show that if the norm of $\sigma $ is finite, then the differential equation extends to every point $x$ on the boundary circle, and the solution to the differential equation an initial condition is unique. The fifth and last part of the paper concerns correspondence of regularity conditions on the measure $\sigma,$ on its corresponding mapping $h_{\sigma},$ and on the tangent vector $$V= V_0 = \frac{d}{dt}\big|_{t=0} h_{t\sigma}.$$ We give equivalent conditions on $\sigma, h_{\sigma}$ and $V$ that correspond to $h_{\sigma }$ being in {\em Diff}$^ {\ 1+\alpha}$ classes, where $0\le \alpha <1$.
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